Theorem lplnneat

Description: No lattice plane is an atom. (Contributed by NM, 15-Jul-2012)

	Ref	Expression
Hypotheses	lplnneat.a	|- A = ( Atoms ` K )
	lplnneat.p	|- P = ( LPlanes ` K )
Assertion	lplnneat	|- ( ( K e. HL /\ X e. P ) -> -. X e. A )

Proof

Step	Hyp	Ref	Expression
1		lplnneat.a	|- A = ( Atoms ` K )
2		lplnneat.p	|- P = ( LPlanes ` K )
3		hllat	|- ( K e. HL -> K e. Lat )
4		eqid	|- ( Base ` K ) = ( Base ` K )
5	4 2	lplnbase	|- ( X e. P -> X e. ( Base ` K ) )
6		eqid	|- ( le ` K ) = ( le ` K )
7	4 6	latref	|- ( ( K e. Lat /\ X e. ( Base ` K ) ) -> X ( le ` K ) X )
8	3 5 7	syl2an	|- ( ( K e. HL /\ X e. P ) -> X ( le ` K ) X )
9	6 1 2	lplnnleat	|- ( ( K e. HL /\ X e. P /\ X e. A ) -> -. X ( le ` K ) X )
10	9	3expia	|- ( ( K e. HL /\ X e. P ) -> ( X e. A -> -. X ( le ` K ) X ) )
11	8 10	mt2d	|- ( ( K e. HL /\ X e. P ) -> -. X e. A )